UDC 517.946.9

MATHEMATICS

L. I. KAMYNIN

ON THE LYAPUNOV–GÜNTER THEOREMS FOR SPECIAL HEAT POTENTIALS

(Presented by Academician S. L. Sobolev on 24 V 1967)

In paper (¹) (see also (², ³)) a theory was constructed for the smoothness of heat potentials of a simple and a double layer with densities distributed on noncylindrical surfaces, entirely analogous to the classical Lyapunov–Günter theory (see (⁴)) for harmonic potentials. In (⁵) (see also (⁶)) the construction of a similar theory was begun for the special heat potentials \(P\) and \(Q\) (introduced by M. Pani in (⁷)), which play an important role in the theory of boundary-value problems with an oblique derivative for a parabolic equation of the second order. The present note, consisting of two sections, contains a complete and systematic investigation in Hölder smoothness spaces of the special heat potential of a simple layer \(P[\varphi]\) and the heat potential \(Q[\varphi]\), which is the derivative of the potential \(P[\varphi]\) in an oblique direction. In § 1 the improving properties of the direct values \(\overline{Q}[\varphi]\) on noncylindrical surfaces of various smoothness are studied, while in § 2 the smoothness of the heat potential \(P[\varphi]\) in the closed domain \(\overline{D}_T\), having \(\Gamma\) as its lateral boundary, is studied. The notation and definitions of the author’s papers (¹, ⁵) are used in the note.

Let \(D_T\) be a bounded domain of the \((n+1)\)-dimensional Euclidean space \((x,t) \equiv (x_1,x_2,\ldots,x_n;t)\), situated between two hyperplanes \(t=0\) and \(t=T>0\) and having an \(n\)-dimensional noncylindrical surface \(\Gamma\) as its lateral boundary. Consider the heat potentials

\[ P(\bar{x},t)\equiv P[\varphi]\equiv \int_0^t d\tau \iint_{\Gamma_\tau} P(\bar{x},t;y,\tau)\varphi(y,\tau)\,d\sigma_y(\tau), \qquad (\bar{x},t)\in D_T, \]

\[ Q(\bar{x},t)\equiv Q[\varphi]\equiv \partial P(\bar{x},t)/\partial \nu(x,t), \qquad (x,t)\in \Gamma_t; \]

\(\Gamma_\tau\) is the section of the surface \(\Gamma\) by the hyperplane \(t=\tau\), on which a field of directions is given with unit vector
\(\nu(x,t)=\{\nu_1(x,t),\ldots,\nu_n(x,t;0)\}\), lying in the section \(\Omega_t\equiv D_T\cap\{t=t\}\) and making an acute angle not exceeding \(\pi/2-d_0\) \((d_0>0)\) with the normal \(N(x,t)\) interior with respect to \(\Omega_t\) at the point \((x,t)\in\Gamma_t\). \(P(\bar{x},t;y,\tau)\) is the special fundamental solution, introduced by M. Pani in (⁷), of the \(n\)-dimensional heat-conduction equation corresponding to the field of oblique directions \(\nu(x,t)\).

§ 1. Improving properties of the direct values of the heat potential \(\overline{Q}[\varphi]\). The notation \(\alpha'\), \(\alpha^0\), \(\alpha^*\), \(\beta'\) from (⁵) will be used.

Theorem 1. Suppose that for \(\Gamma\), \(\nu\), and \(\varphi\) the following conditions are satisfied: \(\Gamma\) is of type

\[ \Pi_{2m+1,\,1,\,(1+\alpha)/2}^{m+1,\,\alpha,\,\alpha/2}, \qquad 0<\alpha\le 1, \]

\[ \nu_j\in H_{2m+1,\,\beta,\,\beta/2}^{m,\,1,\,(1+\beta)/2}(\Gamma), \qquad 0<\alpha\le \beta\le 1, \]

\[ \cos(\nu(x,t),N(x,t))\ge d>0, \qquad (x,t)\in\Gamma_t,\quad d=\mathrm{const},\quad 0\le t\le T, \tag{1} \]

\[ \varphi\in H_{2m-1,\,1,\,(1+\alpha)/2}^{m,\,\alpha,\,\alpha/2}(\Gamma), \]

where

\[ \left|\partial^k\varphi(y,\tau)/\partial\tau^k\right| \le \left|\partial^m\varphi/\partial t^m\right|_{\alpha}\tau^{m-k+\alpha/2}, \]
\[ (y,\tau)\in\Gamma,\qquad k=0,1,2,\ldots,m. \]

Then for \(m=1,2,\ldots\) (for \(m=0\) see Lemma 4 (5)) for \(\overline Q(x,t)\) one has

\[ \overline Q\in H_{2m+1,\alpha,\alpha^*/2}^{m,1,(1+\alpha^*)/2}(\Gamma), \]

where the Hölder constants have the form \((C)|\varphi|_{2m+\alpha}\),

\[ \left|\partial^k\overline Q(x,t)/\partial t^k\right| \le (C)|\varphi|_{2m+\alpha}t^{m-k+(1+\alpha)/2}, \qquad (x,t)\in\Gamma,\qquad k=0,1,\ldots,m, \]

and when

\[ p=0,1,2,\ldots,m,\qquad l_j=0,1,\ldots,2(m-p),\qquad j=1,2,\ldots,k, \]
\[ \sum_{j=1}^{k}l_j=2(m-p), \tag{2} \]

where \(k=n-1\),

\[ \left| \partial^{2m-p+l}\overline Q(x,t)/ \partial t^p\partial x_1^{l_1}\cdots \partial x_{n-1}^{l_{n-1}}\partial x_i^{l} \right| \le (C)|\varphi|_{2m+\alpha}t^{(1+\alpha-l)/2}, \qquad l=0,1. \]

Theorem 2. Suppose that for \(\Gamma\), \(\nu\), and \(\varphi\) the following conditions are satisfied: \(\Gamma\) is of type

\[ \mathcal L_{2m+3,\alpha,\alpha/2}^{m+1,1,(1+\alpha)/2},\qquad 0<\alpha\le 1\quad (m=0,1,2,\ldots), \]

\[ \nu_j\in H_{2m+1,1,(1+\beta)/2}^{m+1,\beta,\beta/2}(\Gamma), \qquad 0<\alpha\le\beta\le 1,\qquad j=1,2,\ldots,m \]

(with (1) satisfied),

\[ \varphi\in H_{2m+1,\alpha,\alpha/2}^{m,1,(1+\alpha)/2}(\Gamma), \]

\[ \left|\partial^{k+l}\varphi(y,\tau)/\partial\tau^k\partial y_i^l\right| \le \left|\partial^m\varphi/\partial t^m\right|_{1+\alpha} \tau^{m-k+(1+\alpha-l)/2} \]

\[ (y,\tau)\in\Gamma,\quad k=0,1,2,\ldots,m;\quad i=1,2,\ldots,n-1;\quad l=0\ \text{for } k<m, \]
\[ l=0,1\ \text{for } k=m. \]

Then for \(m=0,1,2,\ldots\) for \(\overline Q(x,t)\) one has

\[ \overline Q\in H_{2m+1,1,(1+\alpha^*)/2}^{m+1,\alpha^*,\alpha^*/2}(\Gamma), \]

where the Hölder constants have the form \((C)|\varphi|_{2m+1+\alpha}\),

\[ \left|\partial^k\overline Q(x,t)/\partial t^k\right| \le (C)|\varphi|_{2m+1+\alpha}t^{m-k+\alpha/2}, \qquad k=0,1,\ldots,m, \]

and, when (2) is satisfied, where \(k=n-1\),

\[ \left| \partial^{2m+1-p}\overline Q(x,t)/ \partial t^p\partial x_1^{l_1}\cdots \partial x_{n-1}^{l_{n-1}}\partial x_i \right| \le (C)|\varphi|_{2m+1+\alpha}t^{(1+\alpha)/2}, \]

\[ \left( \left| \partial^{2m+1-p}\overline Q(x,t)/ \partial t^{p+1}\partial x_1^{l_1}\cdots \partial x_{n-1}^{l_{n-1}} \right|, \right. \]

\[ \left. \left| \partial^{2m+2-p}\overline Q(x,t)/ \partial t^p\partial x_1^{l_1}\cdots \partial x_{n-1}^{l_{n-1}}\partial x_i\partial x_j \right| \right) \le (C)|\varphi|_{2m+1+\alpha}t^{\alpha/2}. \]

§ 2. Smoothness of the special heat potential of a simple layer \(P[\varphi]\) in the closed domain \(\overline D_T\)

Theorem 3. Suppose that for \(\Gamma\), \(\nu\) the conditions of Theorem 1 are satisfied,

\[ \varphi\in H_{2m+1,\alpha,\alpha/2}^{m,1,(1+\alpha)/2}(\Gamma) \qquad \text{for } m=0,1,2,\ldots, \]

\[ \partial^k\varphi(y,0)/\partial\tau^k\equiv 0, \qquad k=0,1,\ldots,m,\qquad (y,0)\in\Gamma_0, \]

and, when (2) is satisfied, where \(k=n-1\),

\[ \left( \left| \partial^{2m-p+l}\varphi(y,\tau)/ \partial\tau^p\partial y_1^{l_1}\cdots \partial y_{n-1}^{l_{n-1}}\partial y_i^l \right| \le (C)|\varphi|_{2m+1+\alpha}\tau^{(1+\alpha-l)/2}, \qquad l=0,1. \right. \]

Then

\[ P \in H_{2m+1,\,1,\,(1+\alpha)/2}^{m+1,\,\alpha',\,\alpha'/2}(\overline{D}_T), \]

where the Hölder constants have the form \((C)|\varphi|_{2m+1+\alpha}\), and (see (2), where \(k=n\))

\[ \left| \partial^{2m+1+l-p}P(\bar{x},t)/\partial t^p \partial \bar{x}_1^{l_1}\cdots \partial \bar{x}_n^{l_n}\partial \bar{x}_i \right| \leq (C)|\varphi|_{2m+1+\alpha} t^{(1+\alpha-l)/2}, \quad l=0,1, \]

\[ \left| \partial^{2m+1-p}P(\bar{x},t)/\partial t^{p+1}\partial \bar{x}_1^{l_1}\cdots \partial \bar{x}_n^{l_n} \right| \leq (C)|\varphi|_{2m+1+\alpha} t^{\alpha/2}, \quad (\bar{x},t)\in \overline{D}_T \]

(for \(m=0\), Theorem 3 coincides with Lemma 6 of (⁵)).

Theorem 4. Suppose that for \(\Gamma\) and \(\nu\) the conditions of Theorem 2 are fulfilled, and

\[ \varphi \in H_{2m+1,\,1,\,(1+\alpha)/2}^{m+1,\,\alpha,\,\alpha/2}(\Gamma) \quad \text{for } m=0,1,2,\ldots, \]

where

\[ \partial^k \varphi(y,0)/\partial \tau^k = 0, \quad k=0,1,2,\ldots,m+1, \]

and (see (2), where \(k=n-1\))

\[ \left| \partial^{2m+1+l-p}\varphi(y,\tau)/\partial \tau^p \partial y_1^{l_1}\cdots \partial y_{n-1}^{l_{n-1}} \partial y_i \partial y_j \right| \leq (C)|\varphi|_{2m+2+\alpha}\tau^{(1+\alpha-l)/2}, \quad l=0,1; \]

\[ \left| \partial^{2m+1-p}\varphi(y,\tau)/\partial \tau^{p+1} \partial y_1^{l_1}\cdots \partial y_{n-1}^{l_{n-1}} \right| \leq (C)|\varphi|_{2m+2+\alpha}\tau^{\alpha/2}. \]

Then

\[ P \in H_{2m+3,\,\alpha',\,\alpha'/2}^{m+1,\,1,\,(1+\alpha)/2}(\overline{D}_T), \]

where the Hölder constants have the form \((C)|\varphi|_{2m+2+\alpha}\), and, when (2) is fulfilled, where \(k=n\) and \(m\) is replaced by \(m+1\),

\[ \left| \partial^{2(m+1)+l-p}P(\bar{x},t)/\partial t^p \partial \bar{x}_1^{l_1}\cdots \partial \bar{x}_n^{l_n}\partial \bar{x}_i \right| \leq (C)|\varphi|_{2m+2+\alpha}t^{(1+\alpha-l)/2}, \quad l=0,1. \]

The proofs of Theorems 1–4 are carried out by the methods of the papers (², ³, ⁶).

Received
22 V 1967

REFERENCES

¹ L. I. Kamynin, DAN, 160, No. 2, 271 (1965).
² L. I. Kamynin, Differential Equations, 1, No. 6, 799 (1965).
³ L. I. Kamynin, Differential Equations, 2, No. 5, 647 (1966).
⁴ N. M. Günter, The Theory of the Potential and Its Application to the Basic Problems of Mathematical Physics, Moscow, 1953.
⁵ L. I. Kamynin, DAN, 169, No. 4, 761 (1966).
⁶ L. I. Kamynin, Differential Equations, 2, No. 10, 1333 (1966); 2, No. 11, 1484 (1966).
⁷ M. Pagni, Ann. Scuola norm. sup. Pisa, Ser. III, II, fasc. I–II, 73 (1957).